MATH 247
Numerical solutions of
Differential Equations

Instructor: Roberto De Leo.     Term: Spring 2022.

Euler
Cauchy
When: Tuesdays & Thursdays 11am-12:30pm.

Where: my zoom room (email me for the password).

Office: Academic Support Building B, Room 214

Office Hours: just email me anytime and we will arrange for a zoom meeting time.

HU Email: roberto DOT deleo AT howard DOT edu

Preferred Email: roberto AT deleo DOT website

Outline

This course is designed for graduate students of all areas who are interested in numerical methods for differential equations, with focus on concrete implementation of the algorithms in MATLAB and/or Python. Many modern and efficient approaches are presented, after fundamentals of numerical approximation are established. Of particular focus are a qualitative understanding of the considered differential equation, high-order time-stepping approaches, fundamentals of finite difference, finite volume, and finite element methods, and important concepts such as stability, convergence, and error analysis.

Fundamentals: Stability, convergence, error analysis.

Methods: Runge-Kutta, multistep, finite difference, finite volume, finite element methods.

Problems: IVPs, BVPs, Poisson equation and other elliptic problems, advection and transport, diffusion, wave propagation, conservation laws.

Textbooks

Further readings

Videos

Topics to be covered

  • Numerical Solution of Ordinary Differential Equations.
    1. one-step and multi-step methods for non-stiff systems of ordinary differential equations,
    2. extrapolation techniques and automatic step size selection,
    3. one-step and multi-step methods for stiff systems of ordinary differential equations and differential-algebraic equations,
    4. single and multiple shooting techniques, finite difference approximations, and the Ritz-Galerkin method for boundary value problems
  • Numerical Solution of Partial Differential Equations.
    1. Initial/boundary value problems for parabolic and hyperbolic PDEs (one space and one time dimension).
    2. Explicit finite-difference schemes. Implicit finite-difference schemes. Stability.
    3. Parabolic and hyperbolic PDEs in two and three space dimensions.
    4. Boundary value problems for elliptic PDEs.

Tentative Syllabus

  • Week 1: Motivation and introduction to Scientific Computing. Floating point system.
  • Week 2: Unix/Linux, introduction to Python & MATLAB.
  • Week 3 & 4: Initial value problems for ODEs: singlestep and multistep methods, accuracy & stability, explicit vs implicit methods, geometrical integration.
  • Week 5 & 6: Boundary value problems for ODEs: shooting method, finite difference method, Galerkin method.
  • Week 7 & 8: Parabolic PDEs.
  • Week 9 & 10: Hyperbolic PDEs.
  • Week 11 & 12: Elliptic PDEs.
  • Week 13 & 14: Review and writing code for complex problems.

Languages used

  • MATLAB
  • Python
  • Any other language registered students might be interested in.

Grades policy

Grades will be evaluated on the following bases:
  1. (Almost) weekly homework: 50%;
  2. Two take-home midterms: 2x15%;
  3. Final project: 20%.
Based on the total, you will get the following grade: A (90-100%), B (80-89%), C (70-79%), D (60-69%), F (0-59%).

Collaboration

Especially with online classes, teamwork is very important and will give you the occasion to interact with your classmates. You are encouraged in general to work with study partners and also to collaborate on homework. On the other side, you must write your solutions yourself, in your own words, and you must list all collaborators and outside sources of information.
It is a serious violation of academic integrity to copy an answer without attribution.

Academic Code of Student Conduct (please see Howard University handbook)

No copying, unauthorized use of calculators, books, or other materials, or changing of answers or other academic dishonesty will be tolerated. Cheating will not be tolerated. Anyone caught cheating will receive an F for the course and may be expelled from the university.

Grievance Procedure

If you have any problems with the policies or rules of this course, discuss your concerns with your instructor. If you are still unable to come to a satisfactory arrangement, you may contact the Director of Undergraduate Studies, Dr. Jill McGowan, and, if still unsatisfied, the Chair of the Department, Dr. Bourama Toni.

Americans with Disabilities Act

Howard University is committed to providing an educational environment that is accessible to all students. In accordance with this policy, students in need of accommodations due to a disability should contact the Office of the Dean for Special Student Services (202-238-2420, bwilliams@howard.edu) for verification and determination of reasonable accommodations as soon as possible after admission and at the beginning of each semester as needed.

Statement on Sex and Gender-Based Discrimination, Harassment and Violence

Howard's Policy Prohibiting Sex and Gender-Based Discrimination, Sexual Misconduct and Retaliation (aka, the Title IX Policy) prohibits discrimination, harassment, and violence based on sex, gender, gender expression, gender identity, sexual orientation, pregnancy, or marital status. With the exception of certain employees designated as confidential, note that all Howard University employees University - including all faculty members - are required to report any information they receive regarding known or suspected prohibited conduct under the Title IX Policy to the Title IX Office (TitleIX@howard.edu or 202-806-2550), regardless of how they learn of it. For confidential support and assistance, you may contact the Interpersonal Violence Prevention Program (202-836-1401) or the University Counseling Service (202-806-7540). To learn more about your rights, resources, and options for reporting and/or seeking confidential support services (including additional confidential resources, both on and off campus), visit titleix.howard.edu.

COVID-19 STATEMENT

The wearing of a face mask as well as compliance with other health protocols while on campus or in the classroom is mandatory. Students will be directed to leave the classroom if a face mask is not worn properly to cover the nose and mouth. Any student who refuses or fails to comply with the University's requirements and precautions against COVID-19, and any other measures the University advances for the safety and protection of the Howard Community, will constitute a violation of the University's Student Code of Conduct and could result in sanctions up to and including expulsion from the University